Abstract:
A general framework for structural preserving numerical methods
for model reduction by Krylov subspace projections is developed.
The goal is to preserve any substructures of importance in the
matrices $L, G, C, B$ that define the model prescribed by transfer
function $H(s)=L^*(G+s C)^{-1} B$. The framework also works for
eigenvalue problems as model reduction and eigenvalue computation
are deeply related. As an application, quadratic transfer
functions targeted by Su and Craig (J. Guidance, Control, and Dynamics,
14 (1991), pp. 260--267.) is revisited, which leads to an improved
algorithm than Su's and Craig's original one in terms of achieving the
same approximation accuracy with smaller reduced systems. New Gram-Schmidt
type orthogonalization process and new Arnoldi type process that only
orthogonalize the prescribed portion of all basis vectors as opposing
to whole vectors by existing counterparts are also developed. These
new processes are designed as one way to numerically realize the idea
in the general framework.